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I always appreciate feedback!
On Mon, May 25, 2020 at 3:11 PM Lucas Brunno Luna <lucasl...@hotmail.com> wrote:
Dear Jeremy,--
I was aware of EWD1924a, but had not really read it until now; I liked the derivation presented in that paper, and I regard that your JAW114 as an improvement over it for me at least; I especially liked your introduction.
I think it worthwhile to mention that I find that proof very interesting on account of its use of the identity function to formulate the demonstrandum. In former days, if I would at all think about the introduction of a name such as "id" for said concept, I wouldn't know what relevance I should ascribe to it besides the option to rank it as "a waste of characters", as Dijkstra puts it in a response to possible objections to the introduction of the name "skip" for the empty statement. In connection with the later, he remarked that the decimal system was only possible thanks to the introduction of the character "0" for the concept of zero. I am increasingly coming to realize first-hand the value of such nomenclature for concepts such as an "empty number", an "empty command/function" (i.e. one that "leaves things as they are"), etc. and I believe that the presented proof for Cantor's theorem is a good example of the use of the identity function.
Of course, these concepts boil down to the notion of identities in general.
In connection with your conclusion, I think that the colors were a convenient device to help present your development; I don't object to their use if you feel like employing that device.
PS.: I do read those JAWs that you share through this group. My general opinion is that they are usually worthwhile for me to read, and I know that you would like to receive some feedback on them. The trouble is that I usually don't really have anything special to say, so I think it is perhaps better to just remain silent. =(
Em sábado, 23 de maio de 2020 22:03:59 UTC-3, Jeremy Weissmann escreveu:Dear all,It would mean a lot to me if you had a look at the following preview of JAW114 , Heuristics for a proof of Cantor's Theorem . You can find the current version here:This note is based on one by Edsger Dijkstra and Jayadev Misra written in 1999, which you can find here:Let me know what you think!All best,+j
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Haha, that's good to know. Just be aware that I still have a lot to learn, and reading what I could possibly have to say would perhaps not be worth the trouble. As they say: "It is better to remain silent at the risk of being thought a fool, than to talk and remove all doubt of it."
All the best!
Em segunda-feira, 25 de maio de 2020 16:15:05 UTC-3, Jeremy Weissmann escreveu:
I always appreciate feedback!
On Mon, May 25, 2020 at 3:11 PM Lucas Brunno Luna <lucasl...@hotmail.com> wrote:
Dear Jeremy,--
I was aware of EWD1924a, but had not really read it until now; I liked the derivation presented in that paper, and I regard that your JAW114 as an improvement over it for me at least; I especially liked your introduction.
I think it worthwhile to mention that I find that proof very interesting on account of its use of the identity function to formulate the demonstrandum. In former days, if I would at all think about the introduction of a name such as "id" for said concept, I wouldn't know what relevance I should ascribe to it besides the option to rank it as "a waste of characters", as Dijkstra puts it in a response to possible objections to the introduction of the name "skip" for the empty statement. In connection with the later, he remarked that the decimal system was only possible thanks to the introduction of the character "0" for the concept of zero. I am increasingly coming to realize first-hand the value of such nomenclature for concepts such as an "empty number", an "empty command/function" (i.e. one that "leaves things as they are"), etc. and I believe that the presented proof for Cantor's theorem is a good example of the use of the identity function.
Of course, these concepts boil down to the notion of identities in general.
In connection with your conclusion, I think that the colors were a convenient device to help present your development; I don't object to their use if you feel like employing that device.
PS.: I do read those JAWs that you share through this group. My general opinion is that they are usually worthwhile for me to read, and I know that you would like to receive some feedback on them. The trouble is that I usually don't really have anything special to say, so I think it is perhaps better to just remain silent. =(
Em sábado, 23 de maio de 2020 22:03:59 UTC-3, Jeremy Weissmann escreveu:Dear all,It would mean a lot to me if you had a look at the following preview of JAW114 , Heuristics for a proof of Cantor's Theorem . You can find the current version here:This note is based on one by Edsger Dijkstra and Jayadev Misra written in 1999, which you can find here:Let me know what you think!All best,+j
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+ Page 0, section “Nomenclature”, line 3.As far as I know, no one does this in practice and the purpose ofRussell's work is to avoid it and the trouble it leads ---likewise forType Theory and Category Theory, in general.I would advise removing that remark; unless you want to include a bit morecavets for the Axiom of Comprehension.
+ Page 0, “where ϕ does not depend on S”.It seems you're trying to be careful about your definitions and makethe article accessible.Perhaps take care to explain the nature of the ϕ entity ;-)( This is an important concercern, since ϕ is mentioned repeatedly inthe article, without every being told what kind of thing it is. )
+ Page 1, “Dijkstra and Misra’s entire proof is determined by thisfirst design decision, which I adopt here as well, without justification ormotivation. ”This sentence does not instill confidence in the paper.
There are a number of ways to define surjectivity of a function: Pointwisewith ∀-∃ or by counting where points are set, or pointfree via cancellation orvia a pre-inverse. Indeed, “id = f ∘ g” read left-to-right says“every element y of the target is obtainable from f, as witnessed by g(y)”;whence “f is surjective”!⇒ This is what you need on page 3: “(⋯S⋯) = g.S” ;-)
Moreover, the remark on the Axiom of Choice is likely to be unhelpful.In Constructive Mathematics, the two formulations of surjectivityare equivalent **without** such an axiom ;-)
+ As hinted by the above three points, there seems to be a tendency to*both* cover Set Theory comprehensively yet make it accessible.Unfortunately, neither goal is met in full.
+ Page 2, “We can easily ∈ bring into the picture via the Axiom of Extensionality… Now we’ve bridged the first gap, but in doing so created an existentialquantification rather than a universal one.”The axiom does *not* apply: It speaks of equalities, not discrepancies.In ‘applying it’, you actually invoked De Morgan's Law which led to the∃-∀ issue you mention afterwards.The Curse of Knowledge …
We have the Axiom of Extensionality which states that equality of sets is equivalent to equality of their elements:
On May 26, 2020, at 12:30, Lucas Brunno Luna <lucasl...@hotmail.com> wrote:
Oh, yes, Musa's answer has reminded me that I found this sentence on page 0, viz. (emphasis mine)We have the Axiom of Extensionality which states that equality of sets is equivalent to equality of their elements:
somewhat ambiguous, (i.e. initially I thought not that every element in some set X is equal to some element in some set Y and vice-versa, but what amounts to saying that both sets have at most one element because all elements are equal to each other) even though the formula that is introduced just after that makes clear what you mean. In any case, I think it's perhaps better to refrain from using a sentence like that.
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